Region crossing change is an unknotting operation

Region crossing change on spatial-graph diagrams

“…We call such a double point a crossing. A crossing change at a crossing c is a local transformation which changes over/under information of c. A local transformation region crossing change of a knot or link diagram was defined in [5]. We define a region crossing change on a spatial-graph diagram D at a region R of D to be a crossing change at all the crossings on ∂R.…”

“…For example, in Fig. 1, a region crossing change on the region R changes the crossings c 1 , c 2 and c 3 , and a region crossing change on S changes d and e. It was shown in [5] that any crossing change on a knot diagram can be realized by a finite number of region crossing changes. For a link diagram, it was proved in [3] (see also [2]) that any crossing change at a self-crossing of a knot-component, and any pair of crossing changes at non-self-crossings for any two knot-components, can be realized by region crossing changes.…”

When is region crossing change an unknotting operation?

“…Recently, a new local transformation on link diagrams was introduced in [10], named as region crossing change. Here a region crossing change at a region of R 2 divided by a link diagram is defined to be the crossing changes at all the crossing points on the boundary of the region.…”

“…Here a region crossing change at a region of R 2 divided by a link diagram is defined to be the crossing changes at all the crossing points on the boundary of the region. For the case of knots, the theorem below was proved in [10]. Therefore we say region crossing change is an unknotting operation on a link diagram if there exist some regions of R 2 divided by the link diagram such that, if we apply region crossing changes on these regions the new diagram represents a trivial link.…”

“…For example Figure 2 shows the effect of taking region crossing change on the region with capital letter R: #-operation Δ-operation n-gon move Evidently, -operation and n-gon move mentioned above are both special cases of region crossing change. THEOREM 1·1 ( [10]). We remark that during the process Reidemeister moves are forbidden, i.e.…”